MATERIAL VS DISCRETIZATION LENGTH SCALES IN PLASTICITY SIMULATIONS OF SOLID FOAMS

Transcription

1 Rev. Materal Adv. vs Mater. dscretzaton Sc. 35 (13) length scales n astcty smulatons of sold foams 39 MATERIAL VS DISCRETIZATION LENGTH SCALES IN PLASTICITY SIMULATIONS OF SOLID FOAMS Xu Zhang 1,,4, Katerna E. Afants,3,4 and Mchael Zaser 4 1 School of Mechancs and Engneerng, Southwest Jaotong Unversty, Chna Laboratory of Mechancs and Materals, Arstotle Unversty of Thessalonk, Greece 3 Cvl Engneerng-Engneerng Mechancs, Unversty of Arzona, Tucson, AZ, USA 4 Unversty of Erlangen, Department of Materals Scence, Insttute for Materals Smulaton WW8, Receved: August 14, 13 Abstract. The stress-stran response of metal foams has attracted sgnfcant attenton from the mechancs communty due to the local collapse that ndvdual cells experence. Ths collapse results n the development of stran localzaton bands, also known as crushng bands, whch gve rse to a random but localzed accumulaton of damage. Such expermental observatons have been successfully smulated by usng a stran gradent astcty model that s numercally memented through a cellular automaton; stochastcty was accounted for by allowng the ntal yeld stress of each mesh element to vary accordng to a Webull dstrbuton. In the present study new smulatons, along the same lnes, are performed n order to further understand not only the effect of mcrostructure on the stress-stran response of ordered and dsordered foams, but also that of the mesh sze of the smulaton grd. It s found that there are no sgnfcant sze effects n foams of low dsorder, whle such effects exst when a hgher degree of structural dsorder s consdered. 1. INTRODUCTION Metal foams are used n a wde range of apcatons, manly n the automotve and aerospace ndustres, due to ther hgh energy absorpton, hgh specfc rgdty and hgh specfc strength [1,]. For such apcatons t s mportant to ncrease the durablty and safety durng use of the metal foam components, by talorng ther mechancal response, partcularly durng compresson. Expermental studes have revealed that there are usually three deformaton stages durng the compresson of foams [3-6]. Intally, the whole same deforms elastcally, whle after ntal yeldng the cellular structure collapses, gvng a ateau n the stress-stran curve, whch s followed by hardenng. The collapse s ntated by fracture or elastc-astc bucklng of ndvdual foam cells, whch Correspondng author: Katerna E. Afants, e-mal: afants@emal.arzona.edu propagates through the materal by narrow crushng bands. Such type of damage s localzed and cannot be dentfed macroscopcally by observng the stress-stran curves only [7-1]. Ths suggests that the ntrnsc mcrostructural response to compresson must be consdered n order to capture both the mcro and macroscopc mechancal behavor. In dong so, prevous studes [7,11] consdered a gradent astcty framework snce t allows for the mechancal response of the ndvdual foam to nfluence the neghborng cells and therefore, stran localzaton can propagate throughout the materal n the form of bands. To account for the ntaton of collapse n a sngle cell and subsequent band propagaton, as well as for the dsorder that s nherent to metal foams, the ntal yeld stress s taken to vary n each cell, accordng to a Webull

2 4 X. Zhang, K.E. Afants and M. Zaser dstrbuton. To smulate such stochastc effects, the compressve stran gradent astcty model s numercally memented through a cellular automaton (CA). Such smulatons have been able to successfully capture expermentally observed stress-stran curves, and to also predct the dfferences n the mechancal behavor between ordered versus dsordered foams [11]. To further nvestgate the stress-stran response of cellular materals, and n partcular sze effects that result from ther mcrostructure, smulatons are performed heren for dfferent types of foams n whch ether the cell sze or overall specmen sze was vared for both ordered and dsordered foams. When one s concerned wth smulatons, however, the dscretzaton length emoyed n the smulaton, can also result n sze effects, whch are however smulaton artfacts. Therefore we have separately studed the queston of mesh sze dependence and have nvestgated methods to remove mesh sze related artfacts. These nvestgatons provde some parameter analyss for the optmzaton and numercal realzaton of metal foams. In the sequel the stran gradent astcty formulaton s frst presented, followed by the nvestgaton of sze effects. In order to account for smulaton artfacts, frst t wll be verfed that the dscretzaton length does not affect the stress-stran response, and then the effect of the cell sze and specmen sze wll be consdered.. STRAIN GRADIENT PLASTICITY MODEL FOR METAL FOAMS IN COMPRESSION The mass conservaton law and yeld condton should be met by the deformed materal under consderaton. Hence, the evoluton of the metal foam densty s governed by the mass conservaton law as dv( v), (1) e Tr where s the ntal densty of the metal foam and s the stran tensor whose frst nvarant Tr j represents the volume deformaton. Part of the densty s recoverable due to the elastc deformaton, whle the unrecoverable densty resultng from the astc compacton or bucklng (characterzed by the astc stran tensor ) can be wrtten as el Tr Tr e e. () Yeldng and further astc deformaton of the metal foam occurs when the equvalent stress eq s larger than the flow strength S, S, (3) eq where we assume that the ncremental stran s derved from an assocated flow rule. Accordng to the conventonal astcty for metals, deformaton s not assocated wth volume changes, for metal foams however, volumetrc changes ay a crucal role n the ntal deformaton stages and therefore the equvalent stress s defned as [1,13], 3. dev dev eq j j (4) dev The devatorc stress tensor n Eq. (4) s denoted as = j dev j j - 1/3, where summaton s performed over the repeated ndces. The non-dmensonal parameter determnes the volumetrc change that the foam experences durng deformaton and t s chosen to be.5, for consstency wth the expermental observaton that there s no lateral expanson/contracton after unaxal astc deformaton of many alumnum foams. A phenomenologcal expresson [14] for the yeld functon S n Eq. (3) s S S ( ) C, M M (5) where s the stress-free densty (.e. the unrecoverable densty n the unloaded status), and M, M are the yeld stress and densty of the correspondng bulk metal materal. The pre-factor C, whch s related to the foam geometry and astc bucklng, s also further expressed n terms of as C C ( ) C C C 1 e, (6) 1 1 where C corresponds to the ntal strength before compresson, C 1 represents the reduced strength after compresson, and 1 s the characterstc densty change assocated wth astc bucklng. If we ntroduce the parameter = C 1 /C for characterzng the relatve reducton of the strength and the parameter b -1 = 1 / for representng the relatve change of densty, then combnng Eqs. (5), (6), and () gves the fnal expresson for the yeld functon as S S ( ) g(tr ), (7) where S ( ) s the ntal yeld stress that s a functon of the ntal densty, S S ( ) C, M M (8)

3 Materal vs dscretzaton length scales n astcty smulatons of sold foams 41 whle g(tr ) represents the evoluton of the yeld stress and s a functon of the astc stran, Tr g(tr ) e Tr 1 e. (9) Furthermore, to account for the strong nteracton between adjacent cells of the metal foam, a spatal coung term s ntroduced nto the yeld functon through a gradent term [7,15] so that t becomes S S ( ) g(tr ) E l Tr, (1) where s the Laace symbol whch represents the second spatal gradent, E s the elastc modulus, and l s the nternal length scale, whch s nherent to gradent astcty theores and s taken to be related to the cell sze of the metal foam [7,11]. The mcrostructure parameters of metal foams (cell sze, cell shape) are strongly fluctuatng, resultng n spatal fluctuatons of the densty. Snce the ntal yeld strength S s a functon of the densty, t should also fluctuate n space. We descrbe ths fluctuaton usng a Webull dstrbuton wth shape parameter and scale parameter. The correspondng cumulatve dstrbuton functon (CDF) s wrtten as S ;, 1 exp F S (11) and the probablty densty functon (PDF) s gven by 1 S S exp S, f ( S ;, ) (1) S. Thus, the ntal yeld strength can be further wrtten as S =S + S, wth the mean value of S beng S (1 1/ ) and ts varance ( S ) The coeffcent of varaton (COV) s 1 S COV 1. S 1 1 Unaxal compresson of metal foams s a onedmensonal problem. If we assume x as the loadng axs, then xx := and xx = -e are the only non-zero components of the astc stran tensor, thus 3 e eq, Tr. (13) The yeld condton expressed n Eq. (3) can therefore be wrtten as 3 S g( e ) E e. l (14) If we ntroduce the non-dmensonal varables 3 S E, S, E S S S and x/l x, then b e e S e e E e, x x 1. (15) S then obeys a new Webull dstrbuton wth nondmensonal parameters = / S and. Now the cumulatve dstrbuton functon (CDF) s F( S ;, ) 1 exp S and the probablty densty functon (PDF) s (16) 1 S S exp S, f ( S ;, ) (17) S, The correspondng average value equals 1 and the COV remans Eq. (15) s solved usng a cellular automaton method. The whole same wth dmensonless sze of L/l L s dvded nto N sectons/ mesh elements (smlar to the elements n the fnte element method) wth the mesh sze beng = L/N and ts correspondng non-dmensonal verson beng = L /N. The correspondng geometrcal confguraton s shown n Fg. 1. The yeld status s determned by evaluatng Eq. (15) n each dscrete mesh element labeled by

4 4 X. Zhang, K.E. Afants and M. Zaser Fg. 1. The geometrcal confguraton for a metal foam smulated usng a cellular automaton model wth the average cell sze l, mesh length and same sze L. Here the model wth non-dmensonal =3 s schematcally llustrated. ndex. The gradent term n cell s numercally approxmated by the fnte dfference e, x x 1 e e e e e e, x, x e e e. (18) The ntal yeld strength for element follows the Webull dstrbuton, and the correspondng value s evaluated numercally as S R 1/ ln, (19) where R s an equdstrbuted random number rangng from to 1. Accordng to the balance equaton j, =, the axal stress n the compressed metal foam s a constant, whch s equal to the overall stress, and s determned by N 1 E e e. () tot N 1 Substtutng Eqs. (), (19), and (18) nto Eq. (15), provdes the fnal numercal formulaton, whch we use to evaluate the deformaton behavor of the compressed metal foam, as N 1 tot N 1 e ln R e 1 e E e e be 1/ E e e e. 1 1 (1) Ths equaton s the core equaton, whch wll be memented numercally to analyze the mechancal propertes of compressed metal foams. The values for the parameters that are requred to run the nvestgaton of alumnum foams [16]. Partcularly, the ntal elastc modulus and yeld strength are 11 MPa and 5.5 MPa, respectvely, thus the nondmensonal elastc modulus s E =. The compressve behavor of metal foams s always affected tc bucklng occurs when the densty change exceeds 5% of the ntal densty, thus b =, and the correspondng relatve strength reducton s =.47 Durng the smulaton, the total stran s ncreased n small steps from to 1. The step sze s taken to be.4 so as to have a fne resoluton for the otted stress-stran response. In each step, the yeld condton s evaluated for the dscrete mesh elements, and the local astc stran e s ncreased by a small amount of.4 for any element, fulfllng the yeld condton. Then, the global stress s adjusted accordng to Eq. () and the yeld condton (Eq. (1)) s re-evaluated for all mesh elements. Ths procedure s repeated teratvely untl no yeld occurs anywhere. In order to obtan the average stress-stran curve, 5 smulatons are performed and averages are taken. 3. SIZE EFFECT FOR COMPRESSED METAL FOAMS WITH LOW DISORDER For the metal foam wth low dsorder, s taken to be 1. Then the correspondng COV s calculated to be.1. In order to meet the requrement that the mean value of S s 1, s calculated to be 1.5. Usng the aforementoned materals parameters, the effects that the same sze, cell sze, and

5 Materal vs dscretzaton length scales n astcty smulatons of sold foams 43 Fg.. The mesh sze effect for metal foams wth low-dsorder mcrostructure. Fg. 3. The same sze effect for metal foams wth low-dsorder mcrostructure. dscretzaton length (.e. mesh element sze) have on the mechancal response of metal foams are nvestgated as follows: (1) Varyng the mesh sze In the frst case examned, the same sze (L) and the average cell sze (l) were kept constant, L = 6 mm and l =.6 mm, whle the number of mesh elements N was allowed to vary, correspondng to a change of the mesh sze. The parameters used n the smulatons are summarzed n Tables 1&. The correspondng smulaton results for an alumnum metal foam wth low dsorder are summarzed n Fg.. It s seen that the mesh sze does not affect the stress-stran response of low dsorder foams. () Varyng the same sze In the second set of smulatons performed, the average cell sze (l) and the dscretzaton length () were kept constant; l =.6 mm and = 13 mm, respectvely. The specmen sze (L) was allowed to range from 13 mm to 6 mm, n order to nvestgate the effect of same sze. Hence, the number of mesh elements N ncreased wth the specmen length. The dmensonal and, respectve, non-dmensonal parameters characterzng the foams under consderaton are shown n Table &. Fg. 3 llustrates the response of a low dsorder alumnum foam. (3) Varyng the cell sze In the thrd set of smulatons the specmen sze and dscretzaton length were kept constant, L = 5 mm, and = 5., whle the cell sze ranged from.5 mm to 5. mm. The parameters for the nvestgated same are summarzed n Tables 3&. Fg. 4 llustrates the smulaton results for an alumnum metal foam whose mcrostructure exhbts low dsorder for dfferent specmen szes. Fgs. 3 and 4 ndcate that no sze effects exst when the foam mcrostructure s characterzed by low dsorder, regardless of whether the cell sze or same sze was vared. In the metal foam wth low dsorder, the cell sze and the resultng yeld strength were dstrbuted n a narrow band. The ntal yeld strength wth Webull Table 1. The parameters used n the nvestgaton of dscretzaton length effect. wth unt, nondmensonal parameters. L (mm) l (mm) (mm) L N Table. The parameters used n the nvestgaton of specmen sze effect. wth unt, nondmensonal parameters. L (mm) l (mm) (mm) L N

6 44 X. Zhang, K.E. Afants and M. Zaser Table 3. The parameters used the nvestgaton of cell sze effect. wth unt, non-dmensonal parameters. L (mm) l (mm) (mm) Fg. 4. The cell sze effect for metal foams wth lowdsorder mcrostructure. L N parameter = 1 exhbted a comparatvely narrow dstrbuton where the standard devaton of [S ] n Eq. (19) was only.1 of the mean value 1. Thus, the gradent term contrbuton domnated the deformaton behavor, resultng n a propagaton-controlled deformaton mode characterzed by a stress ateau after yeldng. Specfcally, the collapse bands nucleate at the weakest mesh elements and then propagate to the neghborng ones. The propagaton of the bands s asssted by the stran gradent n the band front so as to overcome the hgher yeld stress of the neghborng elements, and the ateau stress, as well as the length of the ateau, are determned by the Maxwell constructon (equal area rule) for the consttutve model of the softennghardenng type as dscussed n [11,17] for analogous problems. After the ateau, deformaton s ndeed homogenous, and compacton contnues at a rapdly ncreasng stress, whch follows the hardenng regme of the softenng-hardenng model ntroduced n Eq. (9). The whole macroscopc stress-stran response of the low dsordered metal foam s smlar to the softenng-hardenng model, snce the gradent term domnates over the dsorder n such cases. The only effect of the dsorder s to facltate band nucleaton and, thus, to elmnate the ntal yeld pont often assocated wth consttutve laws of the softenng-hardenng type. 4. SIZE EFFECT FOR COMPRESSED METAL FOAMS WITH HIGH DISORDER For the metal foam wth hgh dsorder, s taken to be.5; thus the correspondng COV s calculated to be Ths corresponds to foams wth a hghly dsordered, fractal, structure, as the dstrbuton of Fg. 5. The mesh sze effect for metal foams wth hghly dsordered mcrostructure (=.5). fractal mcrostructures wth smlar strength dstrbuton, see [18].) In order to meet the requrement that the mean value S s 1, s calculated to be.4. All other parameters are assumed to have the same values as for the low dsordered metal foam. (1) Varyng the mesh sze In nvestgatng the effect of the dscretzaton length n hghly dsordered metal foams, we consder the case of large sames where the results are ndependent on same sze (cf. () below). We can therefore attrbute the sgnfcant changes n the deformaton behavor seen n Fg. 5 exclusvely to the change n sze of the mesh elements. At frst glance the observaton of such mesh dependence numercal parameter of the model and ts change should not affect the materal behavor. To understand the orgn of ths problem, we need to look at the physcal meanng of the nondmensonal mesh sze: represents how many

7 Materal vs dscretzaton length scales n astcty smulatons of sold foams 45 Fg. 6. Modfcaton of the Webull dstrbuton parameter b and COV to compensate the effect of changng mesh length. The mesh sze effect of hghly dsordered metal foams wth modfed. Fg. 7. The specmen sze effect of metal foams wth hghly dsordered mcrostructure. =.5, =.445. cells are ncluded n one mesh element. By constructon, all the cells n the mesh element share the same ntal yeld stress n the cellular automaton smulaton. If we lump many cells of the physcal mcrostructure nto one element, then the yeld stress of the assembly should have less scatter than f one element contans only a small number of cells. Conversely, f we use elements of dfferent sze, but characterze them wth the same dsorder parameter, then the smulatons wth larger elements actually represent dfferent (more dsordered) mcrostructures than those wth smaller element sze. Accordngly, the results of Fg. 5 ndcate that the latter are weaker than the former. Only n the lmt = 1, the scatter of the mesh element strength dstrbuton drectly reflects the real mcrostructure dsorder and the resultng fluctuatons of ntal yeld stresses. To mtgate ths problem and acheve a representaton of the deformaton behavor that s ndependent on the choce of dscretzaton length, the parameters of the Webull dstrbuton need to be rescaled, together wth the change of the nondmensonal mesh element sze, such as to represent the same deformaton behavor. In ths manner we may elmnate the mesh sze dependence of the macroscopc stress-stran response, whch s also the requrement of a stable smulaton method. The requred change of the Webull dstrbuton parameter wth the change of s shown n Fg. 6a. The new smulaton results shown n Fg. 6b ndcate that the mesh sze dependence can be elmnated by modfyng the dsorder parameter wth the change of when s smaller than 1, whle the mesh sze dependence wll always persst for larger. Thus, there exsts an ntrnsc upper lmt to the scalablty of the model. Ths s ndeed true for all gradent models wth or wthout stochastcty. If aped to stran softenng materals, there s an upper length scale above whch the results must become mesh dependent. For gradent astcty of a materal wth softenng, we get localzed deformaton bands, whch have an ntrnsc wdth that relates to the nternal length scale. For softenng-hardenng we get a propagatng band (see

8 46 X. Zhang, K.E. Afants and M. Zaser Fg. 8. The cell sze effect of metal foams wth hghly dsordered mcrostructure. =.5, after modfcaton of to compensate the mesh sze effect. astc stran profle n [11]) wth a band front wdth w that s agan ntrnsc to the materal parameters. In the smulaton for = 1, the non-dmensonal band wdth (or band front wdth) s about w = w/l = 1, n other words, t s about 1 cell szes. Mesh ndependence means that the band wdth w does not depend on the mesh sze. Qute evdently, a band of wdth w = 1 cannot be represented f the mesh sze s equal to or larger than ths value. As the ntrnsc spatal structure of the deformaton patterns can no longer be represented wth coarser meshng, a scale dependence of the smulaton results s to be expected snce the spatal nteractons between adjacent volume elements can no longer be adequately represented. To summarze: ncreasng reduces the coung between mesh elements (gradent term effect, see Eq. (1)) and therefore makes the materal appear more dsordered -snce the gradent term (whch wants to homogenze deformaton by propagatng a band) now carres less and less weght whle the dsorder remans the same. One can balance ths by reducng the dsorder (ncreasng ), however ths works only as long as one can correctly represent the acton of the gradent term at all,.e. for < 1. Above = 1 the model can no longer correctly represent the stran pattern even wthout dsorder, so dscretzaton artfacts are bound to occur. () Varyng the same sze Now the effect of the specmen sze on the stress-stran response was examned. The parameters lsted n Table were used and the smulaton results are shown n Fg. 7a. The results ndcate that a sze effect exsts when L s smaller than 5. For larger systems (L>5), there s no sze effect. It thus seems that there s a crtcal value of L, above whch the sze effect dsappears. Ths s consstent wth the classcal dea that the behavor of a heterogeneous materal should become scale nvarant above the scale of some dependence of the crtcal value of L on the dsorder parameter, another case wth decreased dsorder (=.445, keepng the mean value at 1) was nvestgated, and the correspondng results are shown n Fg. 7b. It s seen that n ths case a same sze effect exst only when L s smaller than 5, above whch there s no sze effect. Thus, the crtcal value of L whch s ntutvely obvous. Recall also that there s no sze effect n low dsordered metal foams wth = 1. In the sze effect regme (e.g. L = 5, 1, 5, 5 for the stress-stran curves are located at lower stresses for the same stran). The low value of n the Webull dstrbuton of the ntal yeld stress results n a hghly heterogeneous dstrbuton of [S ], whch also vares over a wde range of values. Thus, the ncrease on the number of cells wll ncrease the probablty that cells have very low ntal yeld stresses. Thus, the flow stress n hghly dsordered metal foams decreases as L ncreases. (3) Varyng the cell sze For the nvestgaton of the effect that the cell sze has, the number of mesh elements s kept constant at N = 1, whle the dmensonless mesh sze s allowed to vary. Ths mes that we consder geometrcally smlar specmens: the cell sze and the specmen sze are proportonal to each other. Fg. 8a shows that despte the geometrcal smlarty, changng results n sgnfcant changes n the deformaton behavor. However, these changes are offset once we re-normalze the dsorder to account for the changng non-dmensonal mesh sze

9 Materal vs dscretzaton length scales n astcty smulatons of sold foams 47 accordng to Fg. 6a. Usng the modfed, no sgnfcant cell sze dependence s observed, as shown n Fg. 8b, whch s expected snce proportonal scalng of all mcrostructural parameters should produce materals wth dentcal deformaton behavor. 5. CONCLUSION The effects of varyng the cell sze, specmen sze, and mesh sze n metal foams that are characterzed by low and hgh mcrostructural dsorder, were nvestgated usng a stran gradent astcty framework that was solved numercally usng a cellular automaton. Stochastc effects were accounted for by allowng the ntal yeld stress of the mesh elements to vary accordng to a Webull dstrbuton. For foams that are characterzed by a low dsorder t was seen that the stress-stran curves were not affected ether by the specmen sze or by the mesh length. However, sgnfcant sze effects were obtaned for hghly dsordered foams, when the mesh dependence n the stress-stran response was observed n hghly dsordered foams, whch, however, could be elmnated through the modfcaton of the yeld stress fluctuatons wth the change of the mesh sze. Smlarly, the observed cell sze dependence of the mechancal behavor of dsordered foams, can be elmnated by approprate re-scalng of the yeld stress fluctuatons, reflectng the fact that specmens of dfferent cell sze but smlar densty are geometrcally smlar and should therefore show dentcal deformaton behavor. Ths ponts at an ssue of general mportance for deformaton models whch nclude stochastc fluctuatons nto dscretzed astcty models. In order to prevent such models from dsayng spurous mesh sze dependence of the smulaton results, the stochastc model must be comemented by a scalng rule whch tells how the statstcs depends on the rato between the dscretzaton length and the physcal length scale of the underlyng materal mcrostructure. ACKNOWLEDGEMENT The authors are grateful to KEA Research Councl Startng Grant MINATRAN for ts support. Xu Zhang s grateful to the supports by NSFC (1117), CPSF (13M5345), and the Fundamental Research Funds for the Central Unverstes (SWJTU11CX7). REFERENCES [1] J. Baumester, J. Banhart and M. Weber // Materals & desgn 18 (1997) 17. [] L.J. Gbson and M.F. Ashby, Cellular solds: structure and propertes (Cambrdge unversty press, 1999). [3] H. Bart-Smth, A.F. Bastawros, D.R. Mumm, A.G. Evans, D.J. Sypeck and H.N.G. Wadley // Acta Mater 46 (1998) [4] A.F. Bastawros, H. Bart-Smth and A.G. Evans// J Mech Phys Solds 48 () 31. [5] D. Weare and M.A. Fortes// Advances In Physcs 43 (1994) 685. [6] A.-H. Benoual, L. Froyen, T. Dllard, S. Forest Journal of materals scence 4 (5) 581. [7] K.E. Afants, A. Konstantnds and S. Forest // Journal of Computatonal and Theoretcal Nanoscence 7 (1) 36. [8] S. Forest, J.-S. Blazy, Y. Chastel and F. Moussy // Journal of materals scence 4 (5) 593. [9] M. Doyoyo and T. Werzbck // Int J Plast 19 (3) [1] T. Werzbck and M. Doyoyo // Journal of aped mechancs 7 (3) 4. [11] M. Zaser, F. Mll, A. Konstantnds and K.E. Afants // Materals Scence and Engneerng: A 567 (13) 38. [1] V.S. Deshpande and N.A. Fleck // J Mech Phys Solds 48 () 153. [13] X.H. Zhu, J.E. Carsley, W.W. Mllgan and E.C. Afants // Scr Mater 36 (1997) 71. [14] C. Park and S.R. Nutt // Mater Sc Eng A 99 (1) 68. [15] E.C. Afants // Internatonal Journal of Engneerng Scence (1984) 961. [16] J.-S. Blazy, A. Mare-Louse, S. Forest, Y. Chastel, A. Pneau, A. Awade, C. Grolleron and F. Moussy // Internatonal journal of mechancal scences 46 (4) 17. [17] E.C. Afants and J.B. Serrn // Journal of Collod and Interface Scence 96 (1983) 53. Mater Sc Eng A 7 (1999) 443.